|VW - CVM 1.1|
The coordinates for this lattice are particularly simple: X and Y are simply proportional to x and y. The lattice waves are as in the general lattice , but here we find it convenient to reorganize the typical term as
anmcos(nX) cos(mY) + bnmcos(nX) sin(mY) + cnmsin(nX) cos(mY) + dnmsin(nX) sin(mY)
Among the isometries that will play a role in this analysis, h, d, v, and R are as before . Here, S denotes a reflection about a horizontal line through the center of the cell; Sv is the similar vertical reflections. Likewise, L and Lv are glide reflections with axes through the center of the rectangle. A and B are the mirror and glide parallel to M but a quarter of the way down the cell; when subscripted with v they are a quarter of the way toward the left of the cell.
|The rectangular lattice cell|
Why are these the only candidates for isometries or negating isometries in this case? We give the gist of the idea, leaving rigorous proof for the later algebraic section:
If there are to be reflections at all, they must be parallel to the sides of a rectangular cell or along the diagonal of a rhombic cell. If reflections are too close together, they generate a translation smaller than the ones that already appear in the group.